Question

Design a sine function with the given properties:

Answer 1

Answer:

$f(x)=3\sin\left(\frac{\pi}{12}\left(x-10\right)\right)+13$.

Step-by-step explanation:

Given information:

Period = 24 hr

Maximum = 16 at t=16 hr.

Minimum = 10 at t=4 hr.

The general sin function is

$y=A+\sin(B(x-C))+D$        .... (1)

where, |A| is altitude, $\frac{2\pi}{B}$ is period, C is phase shift and D is midline.

Period is 24 hr.

$24=\dfrac{2\pi}{B}\Rightarrow B=\dfrac{\pi}{12}$

Altitude is

$A=\dfrac{Maximum-Minimum}{2}=\dfrac{16-10}{2}=3$

$D=\dfrac{Maximum+Minimum}{2}=\dfrac{16+10}{2}=13$

The function is minimum at t=4 and maximum at t=16,phase shift is

$C=\dfrac{16+4}{2}=10$

Substitute these values in equation (1).

$y=3\sin\left(\frac{\pi}{12}\left(x-10\right)\right)+13$

Therefore, the required function is $f(x)=3\sin\left(\frac{\pi}{12}\left(x-10\right)\right)+13$.